We integrate Newton's equations using the velocity‑Verlet method:
rᵢ(t+Δt) = rᵢ(t) + vᵢ(t)Δt + ½ aᵢ(t)Δt²
aᵢ = Σⱼ G mⱼ (rⱼ − rᵢ) / (|rⱼ − rᵢ|² + ε²)^{3/2}
vᵢ(t+Δt) = vᵢ(t) + ½[aᵢ(t)+aᵢ(t+Δt)]Δt
It's symplectic (good at conserving energy) but still approximate. Smaller Δt → more accurate. ε avoids singularities when bodies get extremely close.
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